
If \(\sqrt{n}G_n^{(b)}(\hat{\theta }_n)\)  satisfies a conditional CLT then the bootstrap distribution targets the correct asymptotic distribution. The bootstrap is often used to compute standard errors. Consistency of moments further requires uniform integrability.See [1]}. Specifically, to consistently estimate the variance \(V_n\) , the condition requires \(\sup _{n} \mathbb {E}^\star (\Vert \sqrt{n}(\hat{\theta }_n^{(b)} - \hat{\theta }_n)\Vert ^{2+\varepsilon })\)  finite for some \(\varepsilon >0\) . See [2]}, [3]}, [4]}, for applications to moment convergence of the bootstrap in OLS regressions and M-estimation. Here, convergence in distribution of \(\theta _b\)  comes from Proposition REF  and a conditional CLT for \(\theta _b^\star \) , i.e. the growing sum \(\sum _{j=0}^b (1-\gamma )^j [H_n(\hat{\theta }_n)]^{-1}\sqrt{m}G_m^{(b)}(\hat{\theta }_n)\) . Assumption REF  is used to ensure that distributional convergence for \(\sqrt{m}G_m^{(b)}(\hat{\theta }_n)\)  extends to \(\theta _b^\star \) . Primitive conditions are given in Appendix  for \(m\)  out of \(n\)  resampling, and reweighting with Gaussian weights.
